#### Surveys in Approximation Theory, 5 (2010), 165-200.

## Logarithmic Potential Theory with Applications to Approximation Theory

### E. B. Saff

**Abstract**. We provide an introduction to logarithmic potential theory in the complex plane that particularly emphasizes its usefulness in the theory of polynomial and rational approximation. The reader is invited to explore the notions of Fekete points, logarithmic capacity, and Chebyshev constant through a variety of examples and exercises. Many of the fundamental theorems of potential theory, such as Frostmanâ€™s theorem, the Riesz Decomposition Theorem, the Principle of Domination, etc., are given along with essential ideas for their proofs. Equilibrium measures and potentials and their connections with Green functions and conformal mappings are presented. Moreover, we discuss extensions of the classical potential theoretic results to the case when an external field is present.

**E-print:** `arXiv:1010.3760`

Published: 12 October 2010.

- PDF file (374 KB)

Edward B. Saff

Center for Constructive Approximation

Department of Mathematics

Vanderbilt University

Nashville TN 37240 USA

E-mail: edward.b.saff@vanderbilt.edu

Homepage: http://www.math.vanderbilt.edu/~esaff